On the generalized Hurwitz equations and the Baragar-Umeda equations

TL;DR

This paper investigates the solvability of generalized Hurwitz and Baragar-Umeda equations in integers, expanding understanding of their solutions and properties in number theory.

Contribution

It introduces a unified approach to analyze the integer solutions of these two classes of equations, providing new insights and criteria for solvability.

Findings

Derived conditions for integer solutions of the generalized Hurwitz equation

Established criteria for the Baragar-Umeda equation solvability

Connected the properties of these equations to broader number theory concepts

Abstract

We consider the generalized Hurwitz equation $a_1x_1^2+ \cdots +a_nx_n^2 = dx_1 \cdots x_n-k$ and the Baragar-Umeda equation $ax^2+by^2+cz^2=dxyz+e$ for solvability in integers.